Poe leaves it somewhat unclear whether the stone wall would cut off Fortunato's air supply entirely. If so, Fortunato would die in a few minutes, as the answer by clane says. But Montresor wanted a terrible revenge. He would not have wanted his victim to die in a few minutes but to die a long, lingering death. Therefore it seems to me that--even though Poe does not say so--there must have been enough air to keep Fortunato alive. Montresor refers to the dampness and dripping water many many times throughout the story. For example:

“Pass your hand,” I said, “over the wall; you cannot help feeling the nitre. Indeed, it is very damp. Once more let me implore you to return. No? Then I must positively leave you. But I must first render you all the little attentions in my power.”

The wall to which Montresor refers is the one to which his victim is chained. This seems intended to show that Fortunato could get some water but would have to do so by licking it drop by drop off the granite wall. Montresor also refers to the nitre many times. The chemical formula for nitre is KNO3. Without knowing much about chemistry, this seems to me to indicate that the catacombs are full of nitre which contain three parts oxygen. So the place, which as Montresor says is under the river, seems to be full of oxygen and water.

“The nitre!” I said; “see, it increases. It hangs like moss upon the vaults. We are below the river's bed.The drops of moisture trickle among the bones. Come, we will go back ere it is too late. Your cough—”

Poe seems to be taking pains to assure the reader that Fortunato will have enough water and enough foul air. This suggests that Fortunato will starve to death. As clane says in her answer, this means he could stay alive, albeit in total misery, for about three weeks. That would suit Montresor just fine.

Interesting Question. . . From a biological viewpoint there is the rule of 3s. A person can live three minutes without air, three days without water, and three weeks without food. Depending on ventilated the space behind the wall is he may only live a few minutes after the oxygen runs out. If air is coming in, then he could live longer, but he certainly won't make it beyond three weeks. Considering the amount of wine he had drunk prior to being cemented into the space, if there is air, by the time he sobers up fully he will be severely dehydrated so he may not even live three days.